Explicit Teichmüller curves with complemetary series

Abstract

We construct an explicit family of arithmetic Teichmüller curves ${\mathcal{C}}_{2k}$, $k\in \mathbb{N}$, supporting $\text{SL}(2,\mathbb{R})$-invariant probabilities ${\mu }_{2k}$ such that the associated $\text{SL}(2,\mathbb{R})$-representation on $L^2({\mathcal{C}}_{2k},{\mu }_{2k})$ has complementary series for every $k\ge 3$. Actually, the size of the spectral gap along this family goes to zero. In particular, the Teichmüller geodesic flow restricted to these explicit arithmetic Teichmüller curves ${\mathcal{C}}_{2k}$ has arbitrarily slow rate of exponential mixing.

46 pages, 10 figures. Final version (based on the referee report). To appear in Bulletin SMF

References in corpus (2)

• Connected components of the moduli spaces of Abelian differentials with prescribed singularities
• Teichmueller curves, Galois actions and GT-relations

Cited by in corpus (1)

• Some examples of isotropic SL(2,R)-invariant subbundles of the Hodge bundle